# Definition:Limit Point/Topology/Set/Definition 1

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

A point $x \in S$ is a limit point of $A$ if and only if every open neighborhood $U$ of $x$ satisfies:

$A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.